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Astron. Astrophys. 363, 295-305 (2000)

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5. The magnitude and radial gradient of the electron heat flow

Now we evaluate the expression (26) on the basis of the above expressions for [FORMULA] and [FORMULA] given in Eqs. (38) and (39). Furthermore one may realize that [FORMULA] as measure of the velocity dispersion in our parametrized approach simply is a measure of the logarithmic slope of the electron distribution function, i.e. [FORMULA], in just the same way how the electron temperature is determined from the measured electron distribution function by Scime et al. (1994). This suggests clearly that this parameter function [FORMULA] can be set equal to:

[EQUATION]

Thus, we evaluate the expression (26) for the heat conduction flow by:

[EQUATION]

With Eqs. (38) and (39) we can then numerically evaluate expression (43) and obtain with [FORMULA] and [FORMULA] at [FORMULA]AU:

[EQUATION]

which surprisingly enough is just the order of the heat conduction flow found by Scime et al. (1994) (i.e [FORMULA]). It is also consistent with values of between 5.0 to 8.0 [FORMULA] given by Feldman et al. (1975) and Pilipp et al. (1989).

In addition here we are interested in the study of the radial gradient of the electron heat flow which is also measured by Scime et al. (1994) with ULYSSES at its in-ecliptic itinerary to Jupiter. On this in-ecliptic itinerary mission ULYSSES predominantly was embedded in low speed solar wind (see Bame et al. 1993) with an average speed of [FORMULA] km s-1 and average density of [FORMULA]. Evaluating Eq. (38) for these above conditions and setting:

[EQUATION]

we obtain the function [FORMULA] which is displayed as a function of the solar distance r in Fig. 3. As one can see from the linear curve appearing in the double-logarithmic plot of this figure the heat flow [FORMULA] behaves exactly like a power law in the radial coordinate r given by:

[EQUATION]

where the exponent [FORMULA] evaluates to [FORMULA]. This again, is a very nice result since it nearly exactly fits the result derived from ULYSSES solar wind electron observations (see Scime et al. 1994) yielding:

[EQUATION]

[FIGURE] Fig. 3. Shown is the electron heat flow [FORMULA] in units of [FORMULA] as function of the solar distance r for various values of [FORMULA] (i.e 5, 6, 7 AU).

As evident from the additional curves given in Fig. 3 it can be recognized that a variation of the value [FORMULA] plays a very inferior role for the result. Thus it seems as if, with our parametrized solar wind electron distribution function, we do solve two outstanding problems in the thermodynamic behaviour of solar wind electrons at larger distances.

  • 1) The theoretically obtained magnitude of the electron heat flow is much smaller than that expected from the classical Spitzer-Härm theory (Spitzer & Härm 1953) on the basis of a so-called Fourier law with: [FORMULA].

and:

  • 2) The gradient of [FORMULA] obtained from the above theory is larger than expected for a normal collisionless expansion of solar wind electrons ([FORMULA]), but, interestingly enough, is exactly equal to the gradient found by ULYSSES observations (i.e. [FORMULA]).

This also means that in our parametrizing approach it is automatically arranged that free thermal solar wind electron energies are locally dissipated and thus represent a local energy source given by:

[EQUATION]

This energy dissipation is enforced in our approach by the assumption of electron distributions which are truncated Maxwellians all over. In order to maintain such distribution functions in a collisionfree regime some relaxation process must be operative impeding the usual Liouville-Vlasov degeneration of the distribution function. Processes which we consider to be responsible for this relaxation are quasilinear whistler wave - electron interactions which we shall investigate in the next sections.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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